Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $k = \dfrac{21x + 28}{2x} \div \dfrac{27x + 36}{-2} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $k = \dfrac{21x + 28}{2x} \times \dfrac{-2}{27x + 36} $ When multiplying fractions, we multiply the numerators and the denominators. $k = \dfrac{ (21x + 28) \times -2 } { 2x \times (27x + 36) } $ $ k = \dfrac {-2 \times 7(3x + 4)} {2x \times 9(3x + 4)} $ $ k = \dfrac{-14(3x + 4)}{18x(3x + 4)} $ We can cancel the $3x + 4$ so long as $3x + 4 \neq 0$ Therefore $x \neq -\dfrac{4}{3}$ $k = \dfrac{-14 \cancel{(3x + 4})}{18x \cancel{(3x + 4)}} = -\dfrac{14}{18x} = -\dfrac{7}{9x} $